An implicit-explicit (IMEX) extension of the explicit Runge-Kutta-Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme.

Interpolation (acm G.1.1), Ordinary Differential Equations (acm G.1.7), Partial Differential Equations (acm G.1.8)
Stability and convergence of numerical methods (msc 65M12), Method of lines (msc 65M20)
Modelling, Analysis and Simulation [MAS]
Modelling, Analysis and Computation

Verwer, J.G, & Sommeijer, B.P. (2003). An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. Modelling, Analysis and Simulation [MAS]. CWI.